Question

The value of a cell phone decreases exponentially at a rate of 24% each year after its release date. If the initial value of the phone is $700, the situation can be modeled by the equation v(t)=700(0.76)t , where v is the value of the cell phone t years since its release date. Interpret the graph of the equation and identify the true statement. (1 point) Responses As t increases, v increases slowly at first and then quickly. As t increases, v increases slowly at first and then quickly. As t increases, v increases quickly at first and then slowly. As t increases, v increases quickly at first and then slowly. As t increases, v decreases slowly at first and then quickly. As t increases, v decreases slowly at first and then quickly. As t increases, v decreases quickly at first and then slowly. As t increases, v decreases quickly at first and then slowly

Answer 1

To analyze the equation \( v(t) = 700(0.76)^t \), we note that it represents an exponential decay process. In this situation, the initial value of the phone is \( $700 \), and it decreases at a rate of \( 24% \) each year, meaning that \( 76% \) of its value remains each subsequent year.

The graph of this function will show that:

• At \( t = 0 \), the value \( v(0) = 700(0.76)^0 = 700 \).
• As \( t \) increases, the term \( (0.76)^t \) will get smaller because \( 0.76 \) is less than \( 1 \), indicating that the cell phone value \( v(t) \) is decreasing over time.

Since the decrease occurs exponentially, the value of the phone will drop quickly initially and then the rate of decline will slow over time, leading to a gradual leveling off.

Given the context, the true statement regarding the behavior of \( v(t) \) as \( t \) increases is:

As t increases, v decreases quickly at first and then slowly.